Problem 121 - A Complete Resource Book in Chemistry for JEE Main

Question

The reaction \(\mathrm{X} \longrightarrow\) Product follows first-order kinetics, in 40 minutes, the concentration of \(X\) changes from \(0.1 \mathrm{M}\) to \(0.025 \mathrm{M}\), then the rate of reaction when concentration of \(X\) is \(0.01 \mathrm{M}\) is? (a) \(3.47 \times 10^{-5} \mathrm{M} / \mathrm{min}\) (b) \(1.73 \times 10^{-4} \mathrm{M} / \mathrm{min}\) (c) \(1.73 \times 10^{-5} \mathrm{M} / \mathrm{min}\) (d) \(3.47 \times 10^{-4} \mathrm{M} / \mathrm{min}\)

Step-by-Step Solution

Answer

The rate of reaction is \(3.47 \times 10^{-4} \mathrm{M} / \mathrm{min}\), option (d).

Step 1: Understand First-Order Reaction Kinetics

For a first-order reaction, the rate of the reaction is proportional to the concentration of the reactant. The integrated rate law for a first-order reaction is given by \( \ln( \left[ X \right] ) = -kt + \ln( \left[ X_0 \right] ) \), where \( [X] \) is the concentration at time \( t \), \( [X_0] \) is the initial concentration, and \( k \) is the rate constant.

Step 2: Calculate the Rate Constant (k)

We are given that the concentration of \( X \) changes from \( 0.1 \ \mathrm{M} \) to \( 0.025 \ \mathrm{M} \) in 40 minutes. Using the formula from Step 1, substitute and solve for \( k \):\[\ln(0.025) = -k \times 40 + \ln(0.1)\]\[-3.688879 = -40k - 2.302585\]Solving gives:\[k = \frac{-3.688879 + 2.302585}{-40} = 0.0347 \, \mathrm{min}^{-1}\]

Step 3: Calculate the Rate of Reaction

The rate of a first-order reaction is given by \( \text{Rate} = k[X] \). We use \( k = 0.0347 \ \mathrm{min}^{-1} \) and \( [X] = 0.01 \ \mathrm{M} \):\[\text{Rate} = 0.0347 \times 0.01 = 0.000347 \, \mathrm{M} / \mathrm{min}\]This value can also be expressed as:\[3.47 \times 10^{-4} \, \mathrm{M} / \mathrm{min}\]

Step 4: Choose the Correct Option

The calculated rate of reaction when the concentration of \( X \) is \( 0.01 \ \mathrm{M} \) is \( 3.47 \times 10^{-4} \ \mathrm{M} / \mathrm{min} \). Therefore, the correct choice is (d) \( 3.47 \times 10^{-4} \ \mathrm{M} / \mathrm{min} \).

Key Concepts

Rate of Reaction

The rate of reaction refers to how quickly or slowly a reaction occurs. In first-order kinetics, the rate is determined by the concentration of a single reactant. This means that as the reactant is consumed, the rate of the reaction changes. For the exercise, as the concentration of substance \( X \) decreases, the rate at which the reaction proceeds also decreases. Unlike zero-order reactions, the rate in first-order reactions is not constant because it directly depends on the current concentration of the reactant. This is why it is crucial to regularly monitor concentration over time to determine the reaction's rate accurately.

Integrated Rate Law

The integrated rate law for a first-order reaction provides a relationship between the concentrations of reactants and time. It is key in determining the reaction rate over an interval, especially when dealing with decaying reactants. This is represented mathematically as: \[\ln( [ X ] ) = -kt + \ln( [ X_0 ] )\]where

• \( [ X ] \) is the concentration at time \( t \)
• \( [ X_0 ] \) is the initial concentration
• \( k \) is the rate constant

The integrated rate law allows us to calculate how much reactant remains after a given time, which is especially useful when dealing with substances that decrease over time. This is what we used in the exercise to determine how long it takes for the concentration to drop from \( 0.1 \ \mathrm{M} \) to \( 0.025 \ \mathrm{M} \) over 40 minutes.

Rate Constant Calculation

The rate constant \( k \) is a fundamental part of characterizing a reaction's speed in first-order kinetics. To find \( k \), you need both the initial and final concentrations over a specific time interval. In the example provided, we know that the concentration of \( X \) decreased from \( 0.1 \ \mathrm{M} \) to \( 0.025 \ \mathrm{M} \) in 40 minutes. By rearranging the integrated rate law and substituting the known values, we calculate:\[ k = \frac{-\ln( [X]/[X_0] )}{t} = \frac{-(-1.386294)}{40} \]This leads us to find that \( k = 0.0347 \ \mathrm{min}^{-1} \). Understanding the rate constant helps predict how fast a reaction will proceed and adjust conditions accordingly for desired reaction outcomes. It is essential in both predicting reaction behavior and comparing different reactions.